Beat the stage – Play the story, this will unlock as you progress through each stage and beat the final boss.

Score 1,000, 3,000, and 5,000 points – This is easy. Score the points needed for each award. You can farm lairs until they stop spawning creatures then take down the boss. You can re-enter any stage and farm it through again after you’ve beaten it. Be careful while actually battling, If you increase your puzzle score while playing, you’ll get more points per battle. Don’t rush through each fight or you won’t have enough points in the earlier stages.

Someone on the xbox360achievements.org forums posted that this is a good strategy; I take no credit for this:In order to receive easy points each fight, use a monster such as the 2nd and 3rd tier beetles (with Trident). Damage the opponent to low life, and then let your ability gems build up. Make it a priority to remove the enemy ability gems so they don't destroy your own. Simply keep building up until it is impractical to continue, and then match a Trident. Trident will destroy an extremely large area, destroying enemy ability gems, rocks, as well as your other ability gems--setting off a chain reaction of more Tridents, covering most of the field. You should receive a very large overkill bonus(several hundred, I've seen as much as a thousand) as well as receive a large number of puzzle points simply from the buildup period and any cascades or rocks/enemy gems destroyed at the end.

Remove Barrier (A, B, C, D) – These are the wooden barriers located around each stage. It relates to a specific barrier. You can remove these by stepping on the corresponding trigger tile. Once you beat a stage you should go back into it, and as you progress around, you’ll unlock these to continue moving. Some areas are inaccessible until after you’ve cleared the stage boss.

Defeat Monster (A, B, C, D) - These can be distinguished by an icon by the monster's name when encountering them on the stage map. They will typically stand out, by being in dead ends, trapped behind barriers, etc… Similarly to the barriers, you can get these in a second run through of a map/stage.

Defeat Lair of Monsters (A, B, C, D) - When you walk onto a lair on the stage map, you will fight a boss monster that is a bit stronger than the creatures the lair has been producing. Defeating this stronger boss monster will destroy the lair, giving the award. Not all lairs are connected to an award. You’ll probably destroy them all anyways while working towards another award, but I’ll get to that later.

Open Treasure Chest (A, B, C, D) – This is a very uncommon award. In the Grove of Prey stage, there will be chests destroyed by monsters if they reach them before you. You will have to go through the level once or twice to open up all the barriers before you will be able to have a path through to the proper chests. It’s a race in steps to the chest.

Stage 11: ? ? ? ? ? ? ? ? ? ? The Awards won’t show until it has been beaten for the first time (as it’s the last stage of the game). Afterward you will be tasked with defeating the leader of each affinity type, each being an award, as well as a "secret boss". You will need to take the transporters to the sides of the map (only available after completing the stage and going back) to open the barriers. Then in the central area you fought the final boss you will see paths branching out to a number of lairs. Each lair corresponds to a different affinity/beast and award. After clearing all of the affinity lairs and their bosses a new final enemy will spawn where you originally fought the final boss, defeating him will give you the final award for the stage. (Note: You must clear all the lairs in one go to spawn the boss. If you die or leave the level you will have to start over, though the barriers stay down.)

Stage 12:Stage 12 is the Challenger’s Door. This will be rough. I find it is easiest to just replay the first couple of stages while working towards the majority of these Awards, as you don’t want to worry about them as you’re reaching the end.

Overkill - Win a battle with a killing blow that deals more than 10% of your foe's max HP. Probably come naturally.

Not a Scratch - Win a battle with your beast at max HP. Probably come naturally.

Rise to the Challenge - Win a battle using a beast that has less than 20% of its max HP at the start of the battle. Probably come naturally.

Waste Not, Want Not - Win a battle without making a single idle twist(a twist that makes no sets).

These next few will require some planning:

Close Shave - Win a battle while your beast has less than 1% of its max HP. Try with a high HP monster. Try within the first few battles, obviously. Good luck.

Blank Slate - Destroy all the gems on the board with a single twist. Use a Phalanx (Beetle: tier 3). Let Tridents build up on the board. Wait until they are positioned so that the width of their blast will take out every column.

Cascade x 7 - Make a seven cascade with a single twist. Give it up for dumb luck! Gem changing ability gems seem to help a lot. Try to use the Dark Hunter’s (Ant: Tier 3) Nightmare ability which should help with starting more cascades.

7- Gem Match - Make a match of seven or more gems in the same row with a single twist.Hardest. Award. Ever. You will have to repeatedly fight the first Caith Sith and try to setup one of the 7 match designs detailed under "The Knowing" achievement (follow the link to the xbox360achievement.org forum post, it shows a nice way to set up a lot of the different special moves needed for the Cascade x 7, 7-Gem Match and the Perfect Twist x4). Steer clear of a beast that drops Locks.

The Perfect Twist - Make 4 or more sets with a single twist. The wording is just a strange way of saying make a Combo x4. Reference the diagram for Combo Stones under "The Knowing" achievement(follow the link to the xbox360achievement.org forum post, it shows a nice way to set up a lot of the different special moves needed for the Cascade x 7, 7-Gem Match and the Perfect Twist x4). You will most likely get it during normal play by luck. Keep watch for parts of the board with multiple sets of two which could be turned into the right set. If needed, use idle twists to setup the combo if it’s close.

I take only credit of simplifying and reformatting some of the information provided here. A lot of help came from the people/person beyond the link.